Mixtures

[Deepthi Subbu]

Solution A contains 18 ounces of alcohol and 32 ounces of
water. Solution B contains 72 ounces of water and 8 ounces
of alcohol. What percent of Solution B must be added to
Solution A so that the resulting mixture contains 5
7 water?

[Anju@Gurome]

Solution A contains 18 ounces of alcohol and 32 ounces of water. Solution B contains 72 ounces of water and 8 ounces of alcohol. What percent of Solution B must be added to Solution A so that the resulting mixture contains 57 water?

I'm assuming that means 5/7 water.
Hence, ratios of alcohol to water in final solution = 2/5

Let us assume that x parts of B added to A.

Hence, in the final solution,

• Amount of alcohol = (18 + 8x)
  Amount of water = (32 + 72x)

So, (18 + 8x)/(32 + 72x) = 2/5
--> (90 + 40x) = (64 + 144x)
--> (144x - 40x) = (90 - 64)
--> 104x = 26
--> x = 26/104 = 1/4

Hence, required percentage = 100/4% = 25%

[vipulgoyal]

can we do this by allegiation

[GMATGuruNY]

Solution A contains 18 ounces of alcohol and 32 ounces of water. Solution B contains 72 ounces of water and 8 ounces of alcohol. What percent of Solution B must be added to
Solution A so that the resulting mixture contains 5/7 water?

Here's how to solve with ALLIGATION -- a very good way to handle MIXTURE PROBLEMS.
Alligation can be performed only with fractions or percentages.
Thus, the ratios here must be converted.

Step 1: Convert the ratios to FRACTIONS.
A:
Since alcohol:water = 18:32, and 18+32=50, alcohol/total = 18/50 = 9/25.
B:
Since alcohol:water = 8:72, and 8+72=80, alcohol/total = 8/80 = 1/10.
Mixture:
Since water/total = 5/7, alcohol/total= 2/7.

Step 2: Put the fractions over a COMMON DENOMINATOR.
A = 9/25 = 126/350.
B = 1/10 = 35/350.
Mixture = 2/7 = 100/350.

Step 3: Plot the 3 numerators on a number line, with the numerators of A and B (126 and 35) on the ends and the mixture's numerator (100) in the middle.
A 126---------------100---------------35 B

Step 4: Calculate the distances between the numerators.
A 126------26------100------65------35 B

Step 5: Determine the ratio in the mixture.
The ratio of A to B in the mixture is the RECIPROCAL of the distances in red.
A:B = 65:26 = 5:2.

Since A:B = 5:2 = 50:20, and A's actual volume = 50 ounces, the mixture must be composed of 50 ounces of A and 20 ounces of B.
Thus:
(Amount of B in the mixture)/(total volume of B) = 20/80 = 1/4 = 25%.

For two other problems that I solved with alligation, check here:

https://www.beatthegmat.com/ratios-fract ... tml#484583